Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions
We supplement recent results on a class of Bernstein–Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improv...
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irk-123456789-1661812020-02-19T01:26:02Z Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions Gonska, H. Peltenia, R. Статті We supplement recent results on a class of Bernstein–Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improve earlier statements for Bernstein and genuine Bernstein–Durrmeyer operators. Отримані нещодавно результати щодо одного класу операторів Бернштейна-Дуррмейєра, які зберігають лінійні функції, доповнено шляхом вивчення двох граничних випадків та доведення кількісних тверджень типу Вороновської, що містять модулі гладкості першого та другого порядків. Результати узагальнюють та покращують попередні твердження для операторів Бернпггейна та справжніх операторів Бернштейна - Дуррмейєра. 2010 Article Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions / H. Gonska, R.Peltenia // Український математичний журнал. — 2010. — Т. 62, № 7. — С. 913–922. — Бібліогр.: 8 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166181 517.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Gonska, H. Peltenia, R. Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions Український математичний журнал |
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We supplement recent results on a class of Bernstein–Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improve earlier statements for Bernstein and genuine Bernstein–Durrmeyer operators. |
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Gonska, H. Peltenia, R. |
| author_facet |
Gonska, H. Peltenia, R. |
| author_sort |
Gonska, H. |
| title |
Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions |
| title_short |
Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions |
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Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions |
| title_fullStr |
Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions |
| title_full_unstemmed |
Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions |
| title_sort |
quantitative convergence theorems for a class of bernstein–durrmeyer operators preserving linear functions |
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Інститут математики НАН України |
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2010 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/166181 |
| citation_txt |
Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions / H. Gonska, R.Peltenia // Український математичний журнал. — 2010. — Т. 62, № 7. — С. 913–922. — Бібліогр.: 8 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT gonskah quantitativeconvergencetheoremsforaclassofbernsteindurrmeyeroperatorspreservinglinearfunctions AT pelteniar quantitativeconvergencetheoremsforaclassofbernsteindurrmeyeroperatorspreservinglinearfunctions |
| first_indexed |
2025-07-14T20:55:58Z |
| last_indexed |
2025-07-14T20:55:58Z |
| _version_ |
1837657290145005568 |
| fulltext |
UDC 517.5
H. Gonska (Univ. Duisburg-Essen, Germany),
R. Păltănea (Transilvania Univ., Braşov, Romania)
QUANTITATIVE CONVERGENCE THEOREMS FOR A CLASS
OF BERNSTEIN – DURRMEYER OPERATORS
PRESERVING LINEAR FUNCTIONS
ТЕОРЕМИ ПРО КIЛЬКIСНУ ЗБIЖНIСТЬ ДЛЯ ОДНОГО
КЛАСУ ОПЕРАТОРIВ БЕРНШТЕЙНА – ДУРРМЕЙЄРА,
ЯКI ЗБЕРIГАЮТЬ ЛIНIЙНI ФУНКЦIЇ
We supplement recent results on a class of Bernstein – Durrmeyer operators preserving linear functions. This
is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving
the first and second order moduli of smoothness. The results generalize and improve earlier statements for
Bernstein and genuine Bernstein – Durrmeyer operators.
Отриманi нещодавно результати щодо одного класу операторiв Бернштейна – Дуррмейєра, якi зберi-
гають лiнiйнi функцiї, доповнено шляхом вивчення двох граничних випадкiв та доведення кiлькiсних
тверджень типу Вороновської, що мiстять модулi гладкостi першого та другого порядкiв. Результати уза-
гальнюють та покращують попереднi твердження для операторiв Бернштейна та справжнiх операторiв
Бернштейна – Дуррмейєра.
1. Introduction. In the present paper we continue our research on a class of one pa-
rameter operators Uρn of Bernstein – Durrmeyer type which preserve linear functions and
constitute a link between the so-called ”genuine Bernstein – Durrmeyer operators”Un
and the classical Bernstein operators Bn. A predecessor of this paper (see [1]) will ap-
pear soon in the Czechoslovak Mathematical Journal. Investigation on the operators in
question started in a 2007 note by the second author (see [2]). In both articles more
pertinent references can be found. We recall some basic facts.
Denote by LB [0, 1] the space of bounded Lebesgue integrable functions on [0, 1] and
by Πn, the space of polynomials of degree at most n ∈ N0. The following definition
was first given in [2].
Definition 1.1. Let ρ > 0 and n ∈ N, n ≥ 1. Define the operator Uρn : LB [0, 1]→
→ Πn for f ∈ LB [0, 1] and x ∈ [0, 1] by
Uρn(f, x) :=
n∑
k=0
F ρn,k(f) · pn,k(x) :=
:=
n−1∑
k=1
1∫
0
f(t)µρn,k(t)dt
pn,k(x) + f(0)(1− x)n + f(1)xn.
The fundamental functions pn,k are defined by
pn,k(x) =
(
n
k
)
xk(1− x)n−k, 0 ≤ k ≤ n, k, n ∈ N, x ∈ [0, 1].
Moreover, for 1 ≤ k ≤ n− 1,
c© H. GONSKA, R. PĂLTĂNEA, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 913
914 H. GONSKA, R. PĂLTĂNEA
µρn,k(t) :=
tkρ−1(1− t)(n−k)ρ−1
B(kρ, (n− k)ρ)
and
B(x, y) =
1∫
0
tx−1(1− t)y−1dt, x, y > 0,
is Euler’s Beta function. For ρ = 1 we obtain
Un(f, x) = (n− 1)
n−1∑
k=1
1∫
0
f(t)pn−2,k−1(t)dt
pn,k(x)+
+(1− x)nf(0) + xnf(1), f ∈ LB [0, 1],
while, for ρ→∞, for each f ∈ C[0, 1], the sequence Uρn(f, x) uniformly converges to
the Bernstein polynomial
Bn(f, x) =
n∑
k=0
f
(
k
n
)
pn,k(x).
For several further properties of the Uρn the reader is referred to our recent article
[1] from which we will also use some results in the present note.
Here we supplement the results from [1]. In particular, we discuss the case ρ → 0,
consider iterates of Uρn and prove two quantitative Voronovskaya-type assertions, thus
generalizing and improving corresponding earlier results for Un and Bn. Details will be
given below.
2. Previous results on moments. In what follows we write ej(t) = tj , t ∈ [0, 1],
for j ≥ 0. Two basic properties of the functionals F ρn,k are the following:
F ρn,k(e0) = 1, F ρn,k(e1) =
k
n
, 0 ≤ k ≤ n.
This implies
Uρn(e0) = e0, Uρn(e1) = e1,
i.e., the operators Uρn preserve linear function. Clearly this fact has an impact on the
moments and the Voronovskaya-type theorem. In the following we use the definition
Ψ(t) := t(1− t), t ∈ [0, 1].
In [1] we proved the following formulae for the moments of Uρn.
Theorem 2.1. For x, y ∈ [0, 1], we have
Uρn(e0, x) = 1, Uρn(e1 − ye0, x) = x− y,
and, for r ≥ 1,
Uρn((e1 − ye0)r+1, x) =
ρΨ(x)
nρ+ r
(
Uρn((e1 − ye0)r, x)
)′
x
+
+
(1− 2y)r + nρ(x− y)
nρ+ r
Uρn((e1 − ye0)r, x)+
+
rΨ(y)
nρ+ r
Uρn((e1 − ye0)r−1, x).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
QUANTITATIVE CONVERGENCE THEOREMS FOR A CLASS OF BERNSTEIN – DURRMEYER . . . 915
For brevity we use Mr(x) := Mn,r(x) := Mρ
n,r(x) := Uρn((e1 − xe0)r, x), n ≥ 1,
r ≥ 0, x ∈ [0, 1], in what follows. It is immediate that
(Mn,r(x))′ =
(
Uρn((e1 − ye0)r, x)
)′
x
∣∣∣
y=x
− rMr−1(x). (2.1)
Using (2.1) and putting y = x in Theorem 2.1 we obtain the following recursion for
the central moments.
Corollary 2.1.
Mn,0(x) = 1, Mn,1(x) = 0,
and, for r ≥ 1,
Mn,r+1(x) =
r(ρ+ 1)Ψ(x)
nρ+ r
Mn,r−1(x) +
r(1− 2x)
nρ+ r
Mn,r(x) +
ρΨ(x)
nρ+ r
(Mn,r(x))′.
In particular:
Mn,2(x) =
(ρ+ 1)Ψ(x)
nρ+ 1
,
Mn,3(x) =
(ρ+ 1)(ρ+ 2)Ψ(x)Ψ′(x)
(nρ+ 1)(nρ+ 2)
,
Mn,4(x) =
3ρ(ρ+ 1)2Ψ2(x)n
(nρ+ 1)(nρ+ 2)(nρ+ 3)
+
+
−6(ρ+ 1)(ρ2 + 3ρ+ 3)Ψ2(x) + (ρ+ 1)(ρ+ 2)(ρ+ 3)Ψ(x)
(nρ+ 1)(nρ+ 2)(nρ+ 3)
.
3. The case 0 < ρ < 1 revisited. Note that the above equalities for Uρn are true
for 0 < ρ. It is thus of interest to describe the behavior of Uρn as ρ → 0. We show first
that, for any fixed n ≥ 1, Uρn(f ;x) uniformly converges with a certain speed to the first
Bernstein polynomial of f, i.e., to the linear function
B1(f ;x) = f(0)(1− x) + f(1) · x.
To this end we use the following result which essentially comes from the first author’s
dissertation (cf. [3, p. 117]); see also the proof of Theorem 2.1 in [4].
Theorem 3.1. Let L : C[0, 1]→ C[0, 1] be a positive linear operator reproducing
linear functions. Then for f ∈ C[0, 1] and x ∈ [0, 1] the following inequality holds:∣∣L(f ;x)−B1(f ;x)
∣∣ ≤ 9
4
ω2
(
f ;
√
L(e1 · (e0 − e1);x)
)
.
Proof. For g ∈ C2[0, 1] arbitrary we have∣∣L(f ;x)−B1(f ;x)
∣∣ ≤ |(L−B1)(f − g;x)|+ |(L−B1)(g;x)| ≤
≤ (‖L‖+ ‖B1‖)‖f − g‖∞ + |(L−B1)(g;x)| =
= 2‖f − g‖∞ + |(L−B1)(g;x)|.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
916 H. GONSKA, R. PĂLTĂNEA
Since both L and B1 reproduce linear functions, we have
L(B1g) = B1(B1g) = B1g ∈ Π1,
giving ∣∣(L−B1)(g;x)
∣∣ = |L(g;x)−B1(g;x)− L(B1g;x) +B1(B1g;x)| =
= |L(g −B1g;x)| ≤ L(|g −B1g|;x) ≤
≤ 1
2
‖g′′‖∞L(e1(e0 − e1);x).
Thus ∣∣L(f ;x)−B1(f ;x)
∣∣ ≤ 2‖f − g‖∞ +
1
2
‖g′′‖∞ L(e1(e0 − e1);x).
We now use Lemma 2 in [5] (also published in [6]) showing that for 0 < h ≤ 1
2
fixed
and any ε > 0 there is a polynomial p = p(h, ε) such that
‖f − p‖∞ ≤
3
4
ω2(f ;h) + ε,
and
‖p′′‖∞ ≤
3
2h2
ω2(f ;h).
In the above we take g = p and arrive at∣∣L(f ;x)−B1(f ;x)
∣∣ ≤
≤ 3
2
ω2(f ;h) + 2ε+
3
4h2
L(e1(e0 − e1);x)ω2(f ;h).
If L(e1(e0 − e1);x) = 0, then
∣∣L(f ;x) − B1(f ;x)
∣∣ ≤ 3
2
ω2(f ;h) + 2ε for h and ε
arbitrarily small. Hence in this case
∣∣L(f ;x)−B1(f ;x)
∣∣ = 0, and the inequality of the
theorem is true.
Otherwise we take h =
√
L(e1(e0 − e1);x) and let ε tend to zero. This shows that
the inequality is true for all cases of x ∈ [0, 1].
Theorem 3.1 is proved.
It is now easy to derive the following theorem.
Theorem 3.2. For Uρn, 0 < ρ <∞, n ≥ 1, we have
∣∣Uρn(f ;x)−B1(f ;x)
∣∣ ≤ 9
4
ω2
(
f ;
√
nρ− ρ
nρ+ 1
· ψ(x)
)
.
In particular, for any fixed n, we have
lim
ρ→0
Uρnf = B1f
uniformly.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
QUANTITATIVE CONVERGENCE THEOREMS FOR A CLASS OF BERNSTEIN – DURRMEYER . . . 917
Proof. It is only necessary to observe that
Uρn(e1(e0 − e1);x) = Uρn(e1;x)− Uρn(e2;x) =
= x− x2 −
[
Uρn(e2;x)− x2
]
= x(1− x)−Mρ
n,2(x) =
= x(1− x)− ρ+ 1
nρ+ 1
x(1− x) =
nρ− ρ
nρ+ 1
ψ(x).
Theorem 3.2 is proved.
It is also interesting to describe the convergence to f(x) for 0 < ρ < 1 fixed. In
fact, there is uniform convergence for n→∞, however, getting slower and slower as ρ
approaches 0.
Using Corollary 2.2.1 on p. 31 of the second author’s book [7], we have the following
theorem.
Theorem 3.3. For 0 < ρ <∞, n ≥ 1, f ∈ C[0, 1] and x ∈ [0, 1] there holds
∣∣Uρn(f ;x)− f(x)
∣∣ ≤ (1 +
1
2
n(ρ+ 1)
nρ+ 1
)
ω2
(
f ;
√
x(1− x)
n
)
.
Proof. The inequality is trivially true if x ∈ {0, 1}. Otherwise we put h =
=
√
x(1− x)
n
in the theorem cited and arrive immediately at the upper bound claimed.
Theorem 3.3 is proved.
Remark 3.1. The inequality of Theorem 3.3 can also be derived from Theorem
5.2, formula (5.3), case r = 0 in [1]. For a similar inequality see Theorem 2.3, inequal-
ity (2.11) in [2].
In view of
n(ρ+ 1)
nρ+ 1
↗ ρ+ 1
ρ
for n → ∞, the constant in front of ω2(f ; . . .) be-
comes arbitrarily large for ρ close to 0. However, uniform convergence is still warranted
for n→∞. We have seen before that the situation is different for n fixed and ρ→ 0.
Remark 3.2. The linear function B1f is also the uniform limit of over-iterated
operator images [Uρn]mf, if m → ∞. Here n ≥ 1 and 0 < ρ < ∞ are fixed. In fact,
using Corollary 2.4 in [4] it is easy to see that∣∣[Uρn]m(f ;x)−B1(f ;x)
∣∣ ≤
≤ 9
4
ω2
(
f ;
√(
1− ρ+ 1
nρ+ 1
)m
ψ(x)
)
, f ∈ C[0, 1], x ∈ [0, 1].
For n ≥ 1, 0 < ρ < ∞, one has 0 ≤ 1 − ρ+ 1
nρ+ 1
< 1, and this implies uniform
convergence as m→∞.
4. Quantitative Voronovskaya theorem with first order modulus. In the present
section we prove a quantitative Voronoskaja theorem using the least concave majorant
of the first order modulus of continuity. This will be based upon the following general
theorem.
Theorem 4.1 (see [8]). Let q ∈ N0, f ∈ Cq[0, 1] and L : C[0, 1] → C[0, 1] be a
positive linear operator. Then
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
918 H. GONSKA, R. PĂLTĂNEA∣∣∣∣∣L(f, x)−
q∑
r=0
L((e1 − x)r, x)
f (r)(x)
r!
∣∣∣∣∣ ≤
≤ L(|e1 − x|q, x)
q!
ω̃
(
f (q),
1
q + 1
L(|e1 − x|q+1, x)
L(|e1 − x|q, x)
)
.
Here ω̃ is the least concave majorant of the first order modulus of continuity.
We use the above theorem for q = 2, also recalling that Uρn(ei) = ei, i = 0, 1. This
leads to the following theorem.
Theorem 4.2. For Uρn, given as above, f ∈ C2[0, 1], n ≥ 2 and x ∈ (0, 1) we
have ∣∣∣∣nρ+ 1
ρ+ 1
[Uρn(f, x)− f(x)]− 1
2
Ψ(x)f ′′(x)
∣∣∣∣ ≤
≤ 1
2
Ψ(x)ω̃
(
f ′′,
1
3
√
mρ
√
ρ2
(nρ+ 1)2
+
ρΨ(x)
nρ+ 1
)
, (4.1)
where mρ = max
{
5ρ2 + 13ρ+ 12
ρ2
,
(
7ρ2 + 3ρ+ 2
ρ(ρ+ 1)
)2
}
.
Proof. The general estimate from Theorem 4.1 reduces to∣∣∣∣Uρn(f, x)− f(x)− 1
2
Uρn((e1 − x)2, x)f ′′(x)
∣∣∣∣ ≤
≤ 1
2
Uρn((e1 − x)2, x) ω̃
(
f ′′,
1
3
Uρn(|e1 − x|3, x)
Uρn((e1 − x)2, x)
)
.
Using the above representation of the second moment this turns into
∣∣∣∣Uρn(f, x)− f(x)− 1
2
(ρ+ 1)Ψ(x)
nρ+ 1
f ′′(x)
∣∣∣∣ ≤
≤ 1
2
(ρ+ 1)Ψ(x)
nρ+ 1
ω̃
(
f ′′,
1
3
Uρn(|e1 − x|3, x)
Uρn((e1 − x)2, x)
)
. (4.2)
We now consider two cases.
Case 4.1. x ∈
[
ρ
nρ+ 1
, 1− ρ
nρ+ 1
]
.
Using the Cauchy – Schwarz inequality we first observe that
Uρn(|e1 − x|3, x)
Uρn((e1 − x)2, x)
≤
√
Uρn((e1 − x)4, x)
Uρn((e1 − x)2, x)
=
√
M4(x)
M2(x)
.
The above representations of the two moments show that
M4(x)
M2(x)
=
(3ρ(ρ+ 1)n− 6(ρ2 + 3ρ+ 3))Ψ(x) + (ρ+ 2)(ρ+ 3)
(nρ+ 2)(nρ+ 3)
≤
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
QUANTITATIVE CONVERGENCE THEOREMS FOR A CLASS OF BERNSTEIN – DURRMEYER . . . 919
≤ 1
(nρ+ 2)(nρ+ 3)
[
3ρ(ρ+ 1)n+
(ρ+ 2)(ρ+ 3)
Ψ(x)
]
Ψ(x).
In the above interval we have
Ψ(x) ≥ ρ
nρ+ 1
(
1− ρ
nρ+ 1
)
=
ρ[(n− 1)ρ+ 1]
(nρ+ 1)2
.
Therefore
M4(x)
M2(x)
≤ Ψ(x)
(nρ+ 2)(nρ+ 3)
[
3ρ(ρ+ 1)n+
(ρ+ 2)(ρ+ 3)(nρ+ 1)2
ρ(ρn+ 1− ρ)
]
≤
≤ ρΨ(x)
nρ+ 1
[
3(ρ+ 1)
ρ
+ 2
(ρ+ 2)(ρ+ 3)
ρ2
]
=
ρΨ(x)
nρ+ 1
cρ,
where cρ =
5ρ2 + 13ρ+ 12
ρ2
.
Case 4.2. x ∈
[
0,
ρ
nρ+ 1
]
∪
[
1− ρ
nρ+ 1
, 1
]
.
We only consider the left interval; for the right one the statement follows by sym-
metry. We have successively
Uρn(|e1 − x|3, x) = Uρn(2(e1 − x)2(x− e1)+, x) + Uρn((e1 − x)3, x) =
= 2
n−1∑
k=1
x∫
0
(x− t)3µρn,k(t)dt
pn,k(x) + 2(1− x)nx3 +M3(x) ≤
≤ 2x3
n−1∑
k=1
1∫
0
µρn,k(t)dt
pn,k(x) + 2x2Ψ(x) +M3(x) =
= 2x2Ψ(x)
(
1 +
1
1− x
)
+
(ρ+ 1)(ρ+ 2)Ψ(x)Ψ′(x)
(nρ+ 1)(nρ+ 2)
≤
≤ ρ2Ψ(x)
(nρ+ 1)2
[
2 + 2
nρ+ 1
(n− 1)ρ+ 1
+
(ρ+ 1)(ρ+ 2)
ρ2
]
≤
≤ ρ2Ψ(x)
(nρ+ 1)2
dρ,
where dρ = 6 +
(ρ+ 1)(ρ+ 2)
ρ2
(the latter being correct for n ≥ 2). Hence
Uρn(|e1 − x|3, x)
Uρn((e1 − x)2, x)
≤ ρ2Ψ(x)dρ
(nρ+ 1)2
nρ+ 1
(ρ+ 1)Ψ(x)
=
√
ρ2
(nρ+ 1)2
(
ρdρ
ρ+ 1
)2
.
From the above two cases it follows that the r.h.s. in (4.2) is bounded from above by
1
2
(ρ+ 1)Ψ(x)
nρ+ 1
ω̃
(
f ′′,
1
3
√
mρ
√
ρΨ(x)
nρ+ 1
+
ρ2
(nρ+ 1)2
)
,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
920 H. GONSKA, R. PĂLTĂNEA
where mρ = max
{
cρ,
(
ρdρ
ρ+ 1
)2}
. Multiplying both sides by
nρ+ 1
ρ+ 1
gives the final
result.
Remark 4.1. The above multiplication by
nρ+ 1
ρ+ 1
is somewhat arbitrary. It is, for
example, also possible to multiply by nρ+ 1 or simply by n to arrive at slightly different
inequalities.
Remark 4.2. In case ρ = 1 the right-hand side of inequality (4.1) becomes
1
2
Ψ(x)ω̃
(
f ′′, 2
√
1
(n+ 1)2
+
Ψ(x)
n+ 1
)
,
improving the term
1
2
Ψ(x)ω̃
(
f ′′, 4
√
1
(n+ 1)2
+
Ψ(x)
n+ 1
)
,
which is obtained using the approach in [8].
5. Quantitative Voronovskaya theorem with second order modulus. In this sec-
tion we replace the quantity ω̃1(f ′′; ...) by the second order modulus of smoothness
ω2(f ; δ) given by
sup
{
|f(x− h)− 2f(x) + f(x+ h)|, a+ h ≤ x ≤ b− h, 0 ≤ h ≤ δ
}
,
for 0 ≤ δ ≤ 1
2
. The general underlying inequality was recently given in [5] and reads
as follows.
Theorem 5.1. Let L : C[0,1] → C[0,1] be a positive linear operator such that
Lei = ei, i = 0,1. If f ∈ C2[0, 1], then for any 0 < h ≤ 1
2
the following inequality
holds: ∣∣∣∣L(f ;x)− f(x)− 1
2
L((e1 − x)2;x)f ′′(x)
∣∣∣∣ ≤
≤ L((e1 − x)2;x)×
×
{
5
6h
|L((e1 − x)3;x)|
L((e1 − x)2;x)
ω1(f ′′;h) +
(
3
4
+
1
16h2
L((e1 − x)4;x)
L((e1 − x)2;x)
)
ω2(f ′′;h)
}
.
We will use the theorem in its following form.
Corollary 5.1. Putting h =
√
L((e1 − x)4;x)
L((e1 − x)2;x)
and assuming that h > 0, the
inequality in the theorem becomes:∣∣∣∣L(f ;x)− f(x)− 1
2
L((e1 − x)2;x)f ′′(x)
∣∣∣∣ ≤
≤ L((e1 − x)2;x)
{
5
6
|L((e1 − x)3;x)|√
L((e1 − x)2L((e1 − x)4;x)
×
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
QUANTITATIVE CONVERGENCE THEOREMS FOR A CLASS OF BERNSTEIN – DURRMEYER . . . 921
×ω1
(
f ′′;
√
L((e1 − x)4;x)
L((e1 − x)2;x)
)
+
13
16
ω2
(
f ′′;
√
L((e1 − x)4;x)
L((e1 − x)2;x)
)}
.
In the case of Uρn we temporarily write X = Ψ(x), X ′ = Ψ′(x) and
A :=
[M3(x)]2
M2(x)M4(x)
, B :=
M4(x)
M2(x)
.
Explicitly,
A =
(ρ+ 2)2(X ′)2(nρ+ 3)
(nρ+ 2){[3ρ(ρ+ 1)n− 6(ρ2 + 3ρ+ 3]X + (ρ+ 2)(ρ+ 3)}
,
B =
3[ρ(ρ+ 1)n− 2(ρ2 + 3ρ+ 3)]X + (ρ+ 2)(ρ+ 3)
(nρ+ 2)(nρ+ 3)
.
Inserting the information available on Uρn now into the inequality of Corollary 5.1 and
multiplying both sides by nρ+ 1 leads to the assertion of the following theorem.
Theorem 5.2. For Uρn, given as above, f ∈ C2[0, 1], n ≥ 2 and x ∈ [0, 1], the
following inequality holds:∣∣∣∣(nρ+ 1)[Uρn(f, x)− f(x)]− ρ+ 1
2
ψ(x)f ′′(x)
∣∣∣∣ ≤
≤ (ρ+ 1)ψ(x)
{
5
6
√
Aω1
(
f ′′;
√
B
)
+
13
16
ω2(f ′′;
√
B)
}
.
Moreover,
(i) if f ∈ C3[0, 1], then∣∣∣∣(nρ+ 1)[Uρn(f, x)− f(x)]− ρ+ 1
2
ψ(x)f ′′(x)
∣∣∣∣ = ψ(x)O
(
1√
n
)
‖f ′′′‖,
and
(ii) if f ∈ C4[0, 1], then∣∣∣∣(nρ+ 1)
[
Uρn(f, x)− f(x)
]
− ρ+ 1
2
ψ(x)f ′′(x)
∣∣∣∣ = ψ(x)O
(
1
n
)
‖f (4)‖.
In both cases O(. . .) is independent of f and x.
Corollary 5.2. For ρ = 1 this coincides with the inequalities in Theorem 5 of [5].
Proceeding as before, i.e., inserting the information on Uρn into the inequality of
Corollary 5.1, but multiplying both sides by n (instead of nρ+ 1), implies the following
corollary.
Corollary 5.3.
lim
ρ→∞
∣∣∣∣n[Uρn(f ;x)− f(x)
]
− 1
2
n(ρ+ 1)
nρ+ 1
ψ(x)f ′′(x)
∣∣∣∣ ≤
≤ lim
ρ→∞
n(ρ+ 1)
nρ+ 1
ψ(x)
{
5
6
√
Aω1(f ′′;
√
B) +
13
16
ω2(f ′′;
√
B)
}
.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
922 H. GONSKA, R. PĂLTĂNEA
In other words, for n ≥ 2 we have∣∣∣∣n[Bn(f ;x)− f(x)]− 1
2
ψ(x)f ′′(x)
∣∣∣∣ ≤
≤ ψ(x)
{
5
6
|ψ′(x)|√
3(n− 2)ψ(x) + 1
ω1
(
f ′′;
√
3(n− 2)ψ(x) + 1
n2
)
+
+
13
16
ω2
(
f ′′;
√
3(n− 2)ψ(x) + 1
n2
)}
.
This is the same inequality as the one given in Theorem 4 of [5].
1. Gonska H., Păltănea R. Simultaneous approximation by a class of Bernstein – Durrmeyer operators
preserving linear functions // Czech. Math. J. (to appear).
2. Păltănea R. A class of Durrmeyer type operators preserving linear functions // Ann. Tiberiu Popoviciu
Sem. Funct. Equat. Approxim. and Convex. (Cluj-Napoca). – 2007. – 5. – P. 109 – 117.
3. Gonska H. Quantitative Aussagen zur Approximation durch positive lineare Operatoren: Doctor.
Dissertation. – Univ. Duisburg, 1979. – 190 p.
4. Gonska H., Kacsó D., Piţul P. The degree of convergence of over-iterated positive linear operators // J.
Appl. Funct. Anal. – 2006. – 1. – P. 403 – 423.
5. Gonska H., Raşa I. A Voronovskaya estimate with second order modulus of smoothness // Proc. Math.
Inequal. (Sibiu / Romania, Sept. 2008) (to appear).
6. Gonska H., Raşa I. Four notes on Voronovskaya’s theorem // Schriftenr. Fachbereichs Math. Univ.
Duisburg-Essen. – 2009.
7. Păltănea R. Approximation theory using positive linear operators. – Boston: Birkhäuser, 2004.
8. Gonska H. On the degree of approximation in Voronovskaja’s theorem // Stud. Univ. Babeş-Bolyai.
Math. – 2007. – 52. – P. 103 – 115.
Received 12.06.09
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
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